I'm just finishing a math PhD. Talking to CS PhD's my impression is that we do pretty much the same thing, we just work on different domains.
Also, when people ask me, "So What do you actually do studying math?" I reply, "Sit and stare at the wall for several hours a day. Occasionally I write something down."
For systems work you spend a lot of time implementing ideas and running experiments. In a lot of cases it isn't necessarily that difficult to find an interesting idea that's probably viable. The real work is spending a lot of time writing code and work out all the annoying details to get to the point where you have a decent proof of concept.